269 T. H. Havelock. 
plicated form. Numerical calculations are made for both these cases, and 
graphs of the surface elevation are shown in figs. | and 2. 
The second approximation for the circular cylinder is then investigated. 
The first stage is the surface effect due to a doublet at the centre, and the second 
is that due to a distribution of doublets on a certain semicircle. Hxpressions 
can be obtained for the complete surface elevation, but the calculations are 
limited to that part which consists of regular waves to the rear of the cylinder. 
The integrals are investigated and reduced to a form which permits of numerical 
evaluation. Calculations are carried out for various velocities for two different 
cases, namely, when the depth of the centre is twice, and three times, the radius. 
The results are tabulated for comparison, and one may estimate from these 
rather extreme cases the degree of approximation of the first stage. The effect 
of the second stage is to alter both the amplitude and the phase of the regular 
waves. The amplitude of the first-stage waves has a maximum for the velocity 
a/ (gf), where f is the depth of the centre. It appears that the second stage 
increases the amplitude of the waves for velocities less than +/(gf) and decreases 
it for velocities above this value; further, the crests of the waves are moved 
slightly to the rear by an amount which varies with the speed. Some other 
possible applications of the method of images may be mentioned. For a 
doublet in a stream of finite depth, we can take successive images in the bed 
of the stream and in the free surface, and so build up the image system of a 
doubly infinite series of isolated doublets and of line distributions of doublets ; 
this solution may be compared with the direct solution in finite terms which 
may be obtained in this case. Further, similar methods may be used for the 
three-dimensional fluid motion due to a doublet in a stream, and application 
made to the corresponding problem of a submerged sphere. 
Image of Doublet in Stream. 
2. We may either consider the doublet to be at rest in a uniform stream or to 
be moving with uniform velocity in a fluid otherwise at rest ; we choose the 
latter alternative. Take Ox horizontal and in the undisturbed surface of the 
liquid, and Oy vertically upwards. Let the axes be moving with uniform 
velocity c in the direction of Qz, and let there be a two-dimensional doublet 
of moment M at the point (0, —-f) with its axis making an angle « with the 
positive direction of Oz. The velocity potential of the doublet is given by the 
real part of 
Me‘ 
Seah) @) 
266 
